I have the variables $x_{1}, x_{2}, ..., x_{n}$ and the following relations: $$x_{1}\rho_{1}x_{2}$$ $$x_{2}\rho_{2}x_{3}$$ $$...$$ $$x_{n-1}\rho_{n-1}x_{n},$$
where the relations $\rho_{i}\in[=,<,>]$ are known. How to check if the system of inequalities above are consistent or not?
The way you describe the system, it is always solvable, and one solution is given by:
$$x_1=0\\ x_{k+1} = \begin{cases}x_k & \text{ if } \rho_{k} = "="\\ x_k+1 & \text{ if } \rho_{k} = "<"\\ x_k-1 & \text{ if } \rho_{k} = ">"\end{cases}$$
You can also see that you could take any tuple $(a_1,a_2\dots a_{n-1})$ of $n-1$ positive numbers, where $a_k=0$ if $\rho_{k-1}="="$, and define
$$x_1=a_1\\ x_{k+1} = \begin{cases}x_k & \text{ if } \rho_{k} = "="\\ x_k+a_{k+1} & \text{ if } \rho_{k} = "<"\\ x_k-a_{k+1} & \text{ if } \rho_{k} = ">"\end{cases}$$
to get an infinite set of possible solutions.